\newproblem{lay:4_6_19}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 4.6.19}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Suppose the solutions of a homogeneous system of 5 linear equations in 6 unknowns are all multiples of one nonzero solution. Will the system
	necessarily have a solution for every possible choice of constants on the right sides of the equation? Explain.
}{
  % Solution
	The fact that all homogeneous solutions are multiples of one nonzero solution implies that the null space is 1-dimensional. By the Rank Theorem,
	the rank of $A$ (the system matrix) is 5 (so that $5+1=6$). So,
	\begin{center}
		$\mathrm{Rank}\{A\}=\dim\{\mathrm{Col}\{A\}\}=5$
	\end{center}
	Since the matrix $A$ is $5\times 6$, its column space must be a subspace of $\mathbb{R}^5$. On the other side, since its dimension is 5, then
	\begin{center}
		$\mathrm{Col}\{A\}=\mathbb{R}^5$
	\end{center}
	and consequently, for every $\mathbf{b}\in\mathbb{R}^5$ there is a solution of the equation $A\mathbf{x}=\mathbf{b}$.
}
\useproblem{lay:4_6_19}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
